Decomposition in unitary matrices
Conjecture : Any complex or real matrix is the sum of two unitary matrices.
Proof (ideas) :
We know that every complex matrix A could be diagonalized using two unitary matrices U and V : $$ A = UDV^{*} $$ . The matrix D has positive elements : D=diag(d1,…d2) with $$d_1\geq d_2 \geq …\geq d_n \geq 0$$.
A basic result is the following : every diagonal matrix could be diagonalized with n unitary matrix. Indeed, you just have to choose the good coefficient and use the set of matrix {$$ E_i $$} where $$E_i$$ is a diagonal with -1 everywhere, except at position i where there is a +1:
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